arXiv subjects

Uuganbaatar Ninjbat

Publications and source records attributed to Uuganbaatar Ninjbat.

3 recordsLinked to original sources

arXiv2021

The smallest convex $K$-gon containing $N$ congruent disks

Consider the problem of fnding the smallest area convex $k$-gon containing $n\in\mathbb{N}$ congruent disks without an overlap. By using Wegner inequality in sphere packing theory we give a lower bound for the area of such polygons. For several cases where this bound is tight we construct corresponding optimal polygons. We also discuss its solution for some cases where this bound is not tight, e.g. $n = 2$ and $k$ is odd, and $n = 3$; $k = 4$. On the way to prove our results we prove a result on geometric invariants between two polygons whose sides are pairwise parallel, and give a new characterisation for the trisectrix of Maclaurin.

math.OC↗

arXiv2018

A Missing Proof of The Gibbard-Satterthwaite Theorem

A short and direct proof of the Gibbard-Satterthwaite theorem à la Amartya Sen's proof of Arrow's impossibility theorem is given.

math.CO↗

arXiv2018

Side Disks of a Spherical Great Polygon

Take a circle and mark $n\in\mathbb{N}$ points on it designated as vertices. For any arc segment between two consecutive vertices which does not pass through any other vertex, there is a disk centered at its midpoint and has its end points on the boundary. We analyze intersection behaviour of these disks and show that the number of disjoint pairs among them is between $\frac{(n-2)(n-3)}{2}$ and $\frac{n(n-3)}{2}$ and their intersection graph is a subgraph of a triangulation of a convex $n$-gon.

math.CO↗